We can reduce the number of coupling integrals present by using the properties of the KK wavefunctions; namely, the mode equation and completeness relations, Eqs. (REF ) and (REF ). For example, in prior work [1]}, [2]} we showed how the
spin-2 mode equation Eq. (REF ) and the corresponding completeness relation Eq. (REF ) relate some of the \(a\)  and \(b\)  couplings:
\(b_{l^{\prime }m^{\prime }n} = \dfrac{1}{2}\Big [\mu _{l}^{2} + \mu _{m}^{2} - \mu _{n}^{2}\Big ] a_{lmn}\hspace{35.0pt}b_{n^{\prime }n^{\prime }nn} = \dfrac{\mu ^2_n}{3}a_{nnnn}~.\) 
